Suppose $X$ is a smooth projective variety over $\mathbb C$. Then under what conditions, the natural map $H^0(X,mK_X) \otimes H^0(X,nK_X) \to H^0(X,(m+n)K_X)$ for $m, n \in \mathbb{Z}_{>0}$ is surjective?
My case is particularly simple: $X$ is a smooth curve of genus greater than $1$, and I wish $\otimes_{i=1}^m H^0(X, K_X) \to H^0(X, mK_X)$ to be surjective when $m \geq 3$.
However, I was unable to show this or give a conterexample.
I assume you mean $H^0(X, K_X)^{\otimes m}$ rather than $\oplus_{i=1}^m H^0(X, K_X)$. If $X$ is a smooth projective connected complex curve of genus $g \geq 2$, then the map $$H^0(X, K_X)^{\otimes m} \longrightarrow H^0(X, m K_X),$$ is surjective for any $m \geq 0$, as long as $X$ is not hyperelliptic. This is a theorem of M. Noether.
In the case where $X$ is a smooth curve of genus $g\geq 4$, this map fails to be surjective for all $m\geq 2$, for reasons of dimension. On the one hand, Riemann-Roch shows that $h^0(X,mK_X) = (2m-1)(g-1)$ for $m\geq 2$. On the other hand, $\bigoplus_{j=1}^m H^0(X, K_X)$ has dimension $mg$; this less than $(2m-1)(g-1)$ as long as $g\geq 4$, I believe. When $g=3$, the same argument should show that the map is not surjective provided $m\geq 3$.
